Johan Wiesenbauer and J. K.
Andersen got, independently, the next member of this sequence:
7338823. Both also argued that this sequence must be infinite, using the
same basis:

The sequence is clearly infinite. As there are
arbitrary large prime gaps (remember that that for any positive integer n
the numbers n!+2,n!+3,...,n!+n are all composite) it suffices to say that
in order to continue the sequence p_1,..,p_n there is always a prime q
immediately before a sufficiently large prime gap such that all sums
mentioned above are composite. Among all those primes simply choose the
smallest one (Wiesenbauer)

The sequence is infinite. Proof: There are prime
gaps of arbitrary size, e.g. n!+2 to n!+n are all composites for n>1 since
k divides n!+k for k<=n. Let s(t) be the sum of the first t terms in the
sequence. Let p be a prime followed by a prime gap greater than s(t). Then
p satisfies the conditions of the sequence, possibly except minimality.
Either p or a smaller prime is term t+1 in the sequence and the
sequence cannot end (Andersen)